Thibault Hilaire

A Unifying Framework for Finite Wordlength Realizations

A general framework for the analysis of the finite wordlength (FWL) effects of linear time-invariant digital filter implementations is proposed. By means of a special implicit system description, all realization forms can be described. An algebraic characterization of the equivalent classes is provided, which enables a search for realizations that minimize the FWL effects to be made. Two suitable FWL coefficient sensitivity measures are proposed for use within the framework, these being a transfer function sensitivity measure and a pole sensitivity measure. An illustrative example is presented.

Sensitivity-based pole and input-output errors of linear filters as indicators of the implementation deterioration in fixed-point context

Input-output or poles sensitivity is widely used to evaluate the resilience of a filter realization to coefficients quantization in an FWL implementation process. However, these measures do not exactly consider the various implementation schemes and are not accurate in general case. This paper generalizes the classical transfer function sensitivity and pole sensitivity measure, by taking into consideration the exact fixed-point representation of the coefficients. Working in the general framework of the specialized implicit descriptor representation, it shows how a statistical quantization error model may be used in order to define stochastic sensitivity measures that are definitely pertinent and normalized. The general framework of MIMO filters and controllers is considered. All the results are illustrated through an example.

Reliable implementation of linear filters with fixed-point arithmetic

This article deals with the implementation of linear filters or controllers with fixed-point arithmetic. The finite precision of the computations and the roundoff errors induced may have an important impact on the numerical behavior of the implemented system. More-over, the fixed-point transformation is a time consuming and error-prone task, specially with the objective of minimizing the quantization impact. Based on a formalism able to describe every structure of linear filters/controllers, this paper proposes an automatic method to generate fixed-point version of the inputs-to-outputs algorithm and an analysis of the global error added on the output. An example illustrates the approach.

ρ-Direct Form transposed and Residue Number Systems for Filter implementations

This paper deals with a new approach for Infinite Impulse Response (IIR) Filter based on specific structure and arithmetic. The ρ-Direct Form II transposed, introduced by G. Li [1], is an numerically efficient structure for FIR or IIR filters. Compared to classical direct forms, it uses more computations but less bits are necessary for the same precision. These properties made it well conditioned for fixed point arithmetic and its implementations are economical and numerically efficient comparing to other forms. In other hand, Residue Number Systems (RNS) offer an interesting parallelism where operations are made on small values. RNS are well known for improving the performances of DSP filters. We compare our RNS approach to fixed-point implementations of DFI and ρ-DFIIt.

Finite wordlength controller realisations using the specialised implicit form

A specialised implicit state-space representation is introduced to deal with finite wordlength effects in controller implementations. This specialised implicit form provides a macroscopic description of the algorithm to be implemented. So, it constitutes a unifying framework, allowing to encompass various implementation forms, such as the δ-operator, the ρDirect Form II transposed, observer-based and many other realisations usually considered separately in the literature. Different measures quantifying the finite wordlength effects on the overall closed-loop behaviour are defined in this new context. They concern both stability and performance. The gap with the infinite precision case is evaluated classically through the coefficient sensitivity and roundoff noise analysis. The problem of determining a realisation with minimum finite wordlength effects can subsequently be solved using appropriate numerical methods. The approach is illustrated with an example.

Reliable Fixed-Point Implementation of Linear Data-Flows

In this article, we propose a complete methodology to implement a signal processing or control-engineering algorithm described with a linear data-flow into numerical code using fixed-point arithmetic. Our approach is based on a reliable determination of the Worst-Case Peak gain of a filter as well as on rigorous error analysis of roundoff error propagation. It guarantees that no overflow will occur and that the output error due to the finite precision implementation is less than a given bound. Without loss of generality, we consider the linear data-flows given in the form of Simulink block diagram. It is first transposed into an internal matrix-based representation and then the reliable evaluation of the magnitudes of each internal variable is performed. Our approach allows determining the minimum word-length required to achieve a given accuracy. Finally, the methodology is illustrated with numerical examples.

Generalised modal realisation as a practical and efficient tool for FWL implementation

Finite word length (FWL) effects have been a critical issue in digital filter implementation for almost four decades. Although some optimisations may be attempted to get an optimal realisation with regards to a particular effect, for instance the parametric sensitivity or the round-off noise gain, the purpose of this article is to propose an effective one, i.e. taking into account all the aspects. Based on the specialised implicit form, a new effective and sparse structure, named ρ -modal realisation, is proposed. This realisation meets simultaneously accuracy (low sensitivity, round-off noise gain and overflow risk), few and flexible computational efforts with a good readability (thanks to sparsity) and simplicity (no tricky optimisation is required to obtain it) as well. Two numerical examples are included to illustrate the ρ -modal realisations interest.

Reliable Verification of Digital Implemented Filters Against Frequency Specifications

Reliable implementation of digital filters in finiteprecision is based on accurate error analysis. However, a small error in the time domain does not guarantee that the implemented filter verifies the initial band specifications in the frequency domain. We propose a novel certified algorithm for the verification of a filter’s transfer function, or of an existing finite-precision implementation. We show that this problem boils down to the verification of bounds on a rational function, and further to the positivity of a polynomial. Our algorithm has reasonable runtime efficiency to be used as a criterion in large implementation space explorations. We ensure that there are no false positives but false negative answers may occur. For negative answers we give a tight bound on the margin of acceptable specifications.We demonstrate application of our algorithm to the comparison of various finite-precision implementations of filters already fully designed.